How do you solve the inequality #(x^2-2x-24)/(x^2-8x-20)>=0#?

1 Answer
Apr 1, 2018

Answer:

The solution is #x in (-oo,-4] uu (-2, 6] uu (10, +oo)#

Explanation:

Factorise the inequality

#(x^2-2x-24)/(x^2-8x-20)=((x+4)(x-6))/((x+2)(x-10))>=0#

Let #f(x)=((x+4)(x-6))/((x+2)(x-10))#

Perform a sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-4##color(white)(aaaa)##-2##color(white)(aaaaa)##6##color(white)(aaaa)##10##color(white)(aaaa)##+oo###

#color(white)(aaaa)##x+4##color(white)(aaaaa)##-##color(white)(aa)##0##color(white)(aaa)##+##color(white)(aaa)##+##color(white)(aaa)##+##color(white)(aaa)##+###

#color(white)(aaaa)##x+2##color(white)(aaaaa)##-##color(white)(aa)####color(white)(aaaa)##-##color(white)(a)##||##color(white)(a)##+##color(white)(aaa)##+##color(white)(aaa)##+#

#color(white)(aaaa)##x-6##color(white)(aaaaa)##-##color(white)(aa)####color(white)(aaaa)##-##color(white)(a)####color(white)(aa)##-##color(white)(a)##0##color(white)(a)##+##color(white)(aaa)##+#

#color(white)(aaaa)##x-10##color(white)(aaaa)##-##color(white)(aa)####color(white)(aaaa)##-##color(white)(a)####color(white)(aa)##-##color(white)(a)####color(white)(aa)##-##color(white)(a)##||##color(white)(a)##+#

#color(white)(aaaa)##f(x)##color(white)(aaaaaa)##+##color(white)(aa)##0##color(white)(aaa)##-##color(white)(a)##||##color(white)(a)##+##color(white)(a)##0##color(white)(a)##-##color(white)(a)##||##color(white)(a)##+#

Therefore,

#f(x)>=0# when #x in (-oo,-4] uu (-2, 6] uu (10, +oo)#

graph{(x^2-2x-24)/(x^2-8x-20) [-25.65, 25.66, -12.83, 12.84]}